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Coach Newton gets to talking with a fellow passenger on a plane, who says that a math coach “must really be good with numbers.” Coach Newton chuckles and says, “Not really, math is about much more than numbers!” But the remark does provide the inspiration for a couple of number-related problems this week.

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Not Dozens

Chris adds up all of the positive whole numbers less than 1,728 that are not integer multiples of twelve.

What sum does Chris obtain?

Festive Factorial

To ring in the New Year, here’s a problem about the number 2017.

What is the last non-zero digit of 2017! (the factorial of 2017 = 2017 × 2016 × … × 2 × 1)?

Solutions to week 68

Snug Circle. As you can see in the diagram, there are two possible positions for circle a, centered on either point A or point A‘. However, the two positions are symmetric with respect to the y-axis, so the coordinates of point A will have the desired absolute values. The y-coordinate is easy; since radius AC is perpendicular to the x-axis and has length two, the y-coordinate of A is two. To get the x-coordinate, we need the length of horizontal line AD. But that is one leg of a right triangle with other leg BD of length 6 – 2 = 4, and hypotenuse BA of length 6 + 2 = 8. Therefore, the length of AD is √(64 – 16) = √48 = 4√3, and the desired coordinates are (4√3, 2).

Congruence Time. The minute hand on a clock makes a complete circuit of 360° in 60 minutes, so it travels at 6° per minute, and since both minute hands start pointing straight up vertically, each minute hand points 6m degrees clockwise of vertical at m minutes after midnight in Los Angeles. The hour hand makes a complete circuit in 12 hours, which is 720 minutes, and so it moves at half a degree per minute. Thus, the Los Angeles hour hand points m/2 degrees clockwise of vertical at m minutes after midnight. Since the New York clock starts with the hour hand pointing to the three, which is 90° clockwise of vertical, at that same time the New York hour hand points 90 + m/2 degrees clockwise of vertical. When they form the same angle, we must have 6m – m/2 = 90 + m/2 – 6m. Solving, this is 11m = 90, which means that the two clocks make the same angle 8 2/11 minutes after midnight in Los Angeles.